Price-quantity competition with varying toughness∗
Claude d’Aspremont†
Rodolphe Dos Santos Ferreira‡
Received 27 October 2004; Available online 16 March 2008
Abstract
For an industry producing a single homogeneous good, we define and characterize the
concept of oligopolistic equilibrium, allowing for a parameterized continuum of regimes with
varying competitive toughness. This parameterization will appear to be equivalent to the
one used in the empirical literature. The Cournot and the competitive outcomes coincide,
respectively, with the softest and the toughest oligopolistic equilibrium outcome. The concept offers an alternative to the conjectural variations approach with better foundations. It
can be viewed as a canonical description of oligopolistic behavior which can receive different
theoretical justifications and provide a convenient tool for modeling purposes. Two illustrative cases (linear and isoelastic demands) are developed and the possibility of endogenizing
(strategically) the choice of competitive toughness by the firms is examined.
JEL Classification: D43; L13
Key words: Oligopolistic equilibrium; Competitive toughness; Price-quantity competition;
Market share; Market size; Kinked demand; Cournot-Bertrand debate
∗
Games and Economic Behavior, 65, 62–82, 2009. Reprint 2085.
CORE, Université catholique de Louvain, Belgium. CORE is affiliated to ECORE.
‡
BETA-Theme, Université Louis Pasteur, Strasbourg, France and Institut Universitaire de France, France
†
We are grateful to Rabah Amir and Joe Harrington for their useful comments and suggestions. This paper
presents results of the Interuniversity Poles of Attraction Program of the Belgian Science Policy.
1
1
Introduction
A fundamental problem of oligopoly theory is equilibrium indeterminacy. This indeterminacy is
not only of the kind associated with a given equilibrium concept and the possibility of multiple
solutions. It is mainly related to the choice of the equilibrium concept itself. This is most simply
demonstrated in static models by the choice between quantity competition (usually assimilated
to the approach of Cournot, 1838) and price competition (linked to Bertrand’s 1883 critique
of Cournot). However, this second kind of indeterminacy only reflects the variety of observed
competition regimes varying in their degree of toughness and implying different firm conducts.
In order to formalize this variety of oligopoly regimes, we shall adopt an approach that
was pioneered by Shubik (1959), where firms do not privilege one strategic variable but behave
strategically in price-quantity pairs. This approach will lead us to a comprehensive and canonical
concept of oligopolistic equilibrium providing a unified formulation of the whole spectrum of
enforceable non-cooperative equilibria. The proposed concept remains Cournotian and considers
oligopolistic competition as a generalization of monopoly. Two types of competition are at stake:
the struggle of the whole industry for market size, and the struggle of each individual firm for
its market share. With each type of competition will be associated a constraint. The set
of solutions (corresponding to different competition regimes) can be parameterized according
to the relative values of the Lagrange multipliers associated with these two constraints. This
parameterization can be interpretd in terms of competitive toughness. At one extreme, one
finds the Cournot solution as the softest oligopolistic equilibrium, and, at the other extreme,
one finds the competitive equilibrium when competitive toughness is maximal. As mentioned
by Shubik (1959), the price equilibrium (corresponding to Bertrand competition) is a particular
price-quantity equilibrium, which coincides with the competitive equilibrium when all firms are
producing at equilibrium. All other enforceable outcomes are intermediate to these two extremes.
This parameterization will appear to be equivalent to the one used in the empirical literature, building econometric models that incorporate general equations where each firm conduct
in setting price or quantity is represented by a parameter, itself viewed as an index of competi-
2
tiveness. This so-called “conduct parameter method” has been at the basis of the new empirical
industrial organization and has generated a large number of empirical studies (for a synthesis
see Bresnahan, 1989, and, for more recent references, Corts, 1999). It is related to the conjectural variations approach since both methods encompass the same theories of oligopoly. But, as
stressed by Bresnahan, “the phrase ‘conjectural variations’ has to be understood in two ways: it
means something different in the theoretical literature than the object which has been estimated
in the empirical papers” (Bresnahan 1989, p. 1019). Moreover, in the theoretical literature, conjectural variations have been criticized for their lack of theoretical foundations, at least in static
models.1
The goal of the canonical representation of oligopolistic competition that we propose here
is to provide a convenient tool to the theorist (it amounts to use a generalization of Cournot
equilibrium), which is more game-theoretically founded than the conjectural variations theory,
and which, in parallel to the conduct parameter method used by the econometrician, still nests a
continuum of theories of oligopolies. This theoretical tool will be derived and justified from different approaches, such as the “supply-function,” the “facilitating practices” or the “min-pricing
scheme” approaches.2 Each of these approaches may be more relevant for specific industries. For
example, introducing meeting-the-competition clauses (as a facilitating practice) cane appropriate for retail (e.g. Sears catalogue) and intermediate product markets, or using supply-function
models can be fruitful, for example, in studying the electric power generation market.3 Our
canonical representation is meant for all and readily amenable to empirical testing.
In Section 2 we define and characterize the concept of oligopolistic equilibrium for an industry
producing a single homogeneous good. The resulting parameterization is derived and compared
to the empirical conduct parameterization. In Section 3, we examine some alternative theoretical
justifications of the concept. In Section 4, we analyze the two standard cases of a linear and of
an isoelastic demand and, in Section 5, we examine the possibility of endogenizing (strategically)
the choice of competitive toughness by the firms. We briefly conclude in Section 6.
1
For references and a discussion, see Martin (2002, pp. 50–51).
See, respectively, Grossman (1981), Salop (1986), and d’Aspremont et al. (1991).
3
See, for example, Green and Newbery (1992).
2
3
2
Oligopolistic equilibrium
We take an industry consisting of n firms (n > 1), each firm i producing the same good with a
technology described by an increasing cost function Ci , which is continuously differentiable on
(0, ∞) and such that Ci (0) = 0. The demand D for the good is a function of market price P ,
with a finite continuous derivative D0 (P ) < 0 over all the domain where it is positive and such
that limP →P D (P ) = 0, for some P ∈ (0, ∞]. More assumptions on costs and demand will be
introduced along the way.
2.1
Definition and characterization
In our approach, neither the price nor the quantity will be privileged as a strategic variable.4
Each firm i is supposed to choose, simultaneously with its competitors, a price-quantity pair
(pi , qi ) ∈ IR2+ . We introduce the concept of oligopolistic equilibrium by combining the Cournot
and the Bertrand approaches. The following firm conduct can be assumed. As in the standard
interpretation of Cournot competition, qi is the quantity of output that firm i decides to supply.
This means that the production in advance assumption used by Maskin (1986) holds. However,
we will not consider Bertrand-Edgeworth competition but pure Bertrand competition as defined,
for instance, in Vives (1999, p. 117): pi is the list price at which firm i is ready to supply all
demand, whenever this price is lowest. If more than one firm list the lowest price a sharing rule
has to be applied to allocate the excess demand. This is given by functions s1 , s2 , · · · , sn defined
on IR2n
+ such that:
si (p, q) > 0 if i ∈ arg min and
j
si (p, q) = 0 otherwise,
and
X
D min{pj } >
qj ,
j
j
X
X
si (p, q) = D min{pj } −
qj .
i
4
j
(1)
j
See Shubik (1959) and Shubik and Levitan (1980). Friedman (1988), who investigates differentiated-products
oligopoly, mentions a homogeneous-goods duopoly model of Alger (1979). Closest to the following definition is
Maskin (1986).
4
Because of each firm commitment to serve all demand, no consumer will accept to pay more
than the lowest price min{p1 , · · · , pn }. Moreover, because of production in advance, any firm
is willing to sell all its output at the discount price P = min{p1 , · · · , pn , D−1 (q1 + · · · + qn )}.
Therefore firm i payoff can be defined as:





X
−1 
CB

[q + si (pi , p−i , qi , q−i )]
qi +
qj
Πi (pi , p−i , qi , q−i ) ≡ min pi , p−i , D
 i

j6=i
(2)
−Ci (qi + si (pi , p−i , qi , q−i )).
We thus obtain a Cournot-Bertrand oligopoly game in quantities and prices. An oligopolistic
equilibrium is a 2n-tuple (p∗ , q ∗ ) which is a Nash equilibrium of this oligopoly game that satisfies
in addition a credibility condition: at equilibrium, no firm should be forced to sell more than it
would voluntarily sell, i.e. s(p∗ , q ∗ ) = 0 or, equivalently,
X
∗
∗
qj = D min{pj } .
(3)
j
j
This credibility condition eliminates equilibria where the commitment to serve all demand is
binding for some firm. Notice that this definition does not exclude the case where at equilibrium
a firm i chooses the quantity qi∗ = 0, that is, to be inactive.
The following lemma delivers the main tool of this paper. It gives a formal and canonical
characterization of the concept of oligopolistic equilibrium which is more convenient to use for
modeling purposes and leads, as we will see, to the standard econometric parameterization of
the set of equilibria.
Lemma 1 A 2n-tuple (p∗ , q ∗ ) is an oligopolistic equilibrium if and only if, for each firm i,
(p∗i , qi∗ ) is solution to the program,





X
max
pi qi − Ci (qi ) : pi ≤ min{p∗j } and pi ≤ D−1 qi +
qj∗  ,

j6=i
(pi ,qi )∈IR2+ 
(4)
j6=i
and satisfies
X
qj∗
=D
j
5
min{p∗j }
j
.
(5)
Proof:
Let us start by proving sufficiency. Suppose, that, for each i, (p∗i , qi∗ ) is solution to
(4), satisfying (5), but that, for some i, and some (pi , qi ) ∈ IR2+ ,





X
∗
∗
min pi , p∗−i , D−1 qi +
qj∗  [qi + si (pi , p∗−i , qi , q−i
)] − Ci (qi + si (pi , p∗−i , qi , q−i
))


j6
=
i





X
qj∗  qi∗ − Ci (qi∗ ).
> min p∗i , p∗−i , D−1 qi∗ +


j6=i
P
∗ ) = 0 and the pair (p0 , q ), with
If D−1 (qi + j6=i qj∗ ) ≤ min{pi , p∗−i }, then si (pi , p∗−i , qi , q−i
i i
P
p0i = D−1 (qi + j6=i qj∗ ), satisfies the two constraints of (4), giving higher profit to firm i, which is
P
a contradiction. If D−1 (qi + j6=i qj∗ ) > min{pi , p∗−i }, then the pair (p0i , qi0 ), with p0i = min{pi , p∗−i }
∗ ), again satisfies the two constraints of (4), with a higher profit
and qi0 = qi + si (pi , p∗−i , qi , q−i
for firm i, leading to a contradiction. It remains to observe that the credibility condition (3) is
automatically satisfied since it coincides with (5).
To prove necessity, suppose that, for some i, and some (pi , qi ) ∈ IR2+ , s.t. pi ≤ minj6=i {p∗j }
P
P
and pi ≤ D−1 (qi + j6=i qj∗ ), pi qi − Ci (qi ) > P ∗ qi∗ − Ci (qi∗ ), with P ∗ = minj {p∗j } = D−1 ( j qj∗ ),
and (p∗ , q ∗ ) an oligopolistic equilibrium. WLOG we may suppose that at least one of the two
P
constraints holds as an equality. If pi = D−1 (qi + j6=i qj∗ ), demand is still served with the pricequantity pair (pi , qi ), thus generating a profitable deviation in the Cournot-Bertrand oligopoly
P
game. If pi = minj6=i {p∗j } < D−1 (qi + j6=i qj∗ ), any price-quantity pair (p0i , qi ), with p0i > pi ,
leads to the payoffpi qi − Ci (qi ), since pi = minj6=i {p∗j } < min{p0i , D−1 (qi + /sumj6=i qj∗ )} and
∗ ) = 0, so that it is a profitable deviation in the oligopoly game. Finally, observe
s(p0i , p∗−i , qi , q−i
that (5) coincides with the credibility condition (3).
From now on, on the basis of this lemma, we shall call oligopolistic equilibrium any 2n-tuple
(p∗ , q ∗ ) satisfying (5) and such that, for each i, (p∗i , qi∗ ) is solution to (4). It has been derived as an
equilibrium of the Cournot-Bertrand oligopoly game, but will receive other justifications below.
In fact, it can also receive a direct interpretation. Indeed the profit maximization program (4)
includes the two types of competition in which a firm is involved: the one of the whole industry
for market size (represented by the aggregate demand constraint of the Cournot kind), and the
6
one of each individual firm for its market share (represented by the constrained imposed by the
minimal price).
In order to further justify this characterization of an equilibrium, we start by comparing the
set of oligopolistic equilibria with standard oligopolistic outcomes. We let (P C , q C ) denote a
Cournot outcome, that is, a quantity vector q C satisfying:






X
qiC ∈ arg max
D−1 qi +
qjC  qi − Ci (qi ) ,

qi ∈[0,∞) 
i = 1, · · · , n,
(6)
j6=i
and a price P C
P
= D−1 ( j qjC ). Alternatively, a competitive (Walrasian) outcome is denoted
(P W , q W ) and is such that
qiW ∈ arg max {P W qi − Ci (qi )},
qi ∈[0,∞)
with P W satisfying
W
j qj
P
i = 1, · · · , n,
(7)
= D(P W ). As far as Bertrand competition is concerned, we only
consider, for simplicity, the case where firms have constant marginal costs5 : Ci (qi ) = ci qi ,
i = 1, · · · , n. Then, a Bertrand outcome (P B , q B ) with P B = mini {pB
i } is characterized by
B
pB
i ∈ arg max (pi − ci )di (pi , p−i )
(8)
pi ∈[0,∞)
D(min{pi , pB
−i })
B
, if pi = min{pi , pB
−i }, with di (pi , p−i )
B
# arg min{pi , p−i }
= 0, otherwise.
=
Finally, a collusive outcome (P m , q m ) corresponds to
(
)
X
X
X
(P m , q m ) ∈ arg max
P
qi −
Ci (qi ) :
qi ≤ D (P ) .
(P,q)∈IRn+1
+
5
i
i
(9)
(10)
i
We do not exclude the case where constant marginal costs differ from one firm to another. However, in that
case, to avoid non-existence, it can be assumed that profits are measured in (small) discrete units of account (e.g.
in cents). We then compute the corresponding equilibrium prices and take their limit as the size ε of the unit
of account tends to zero. In the duopoly case, for example, taking c ∈ (c1 , c2 ] and a positive integer mε to be
such that c1 ≤ εmε ≤ c ≤ ε(mε + 1), the prices p1 = εmε and p2 = ε(mε + 1) form an equilibrium (where only
firm 1 produces), and varying c we get all “Bertrand equilibrium” outcomes. See, for example, Mas-Colell et
al. (1995), Exercise 12.C.4. Another way to get all Bertrand equilibria is to have firm 2 randomizing in a small
enough interval above firm 1 price (see Blume, 2003).
7
We can now state the following:
Proposition 2 Any Cournot outcome (P C , q C ), competitive outcome (P W , q W ) and (when
marginal costs are constant) any Bertrand outcome (P B , q B ) is an oligopolistic equilibrium outcome, but a collusive outcome (P m , q m ) is not (unless it is also a Cournot outcome).
If (P r , q r ) (with r = C, W, B) were not an oligopolistic equilibrium outcome, some i
P
would be able by (4), through some choice (pi , qi ) ∈ IR2+ s.t. pi ≤ min{P r , D−1 (qi + j6=i qjr )}, to
P
get a profit pi qi − Ci (qi ) > P r qir − Ci (qir ). This implies P C qiC − Ci (qiC ) < D−1 (qi + j6=i qjC )qi −
Proof:
Ci (qi ) in the Cournot case, contradicting the assumption that q C is a Cournot equilibrium. In
the competitive case, we get P W qiW −Ci (qiW ) < P W qi −Ci (qi ), resulting again in a contradiction.
An in the Bertrand case, the same argument holds directly if min{pi , P B } = P B . Otherwise,
with pi wP B, the deviation (pi , qi ) would a fortiori be feasible in the Bertrand game, where i
would be constrained by the aggregate demand (instead of the residual demand, according to the
P
second constraint in (4), that is qi ≤ D(pi ) − j6=i qiB ). Finally, if (P m , q m ) is a collusive but not
P
a Cournot outcome, it must be such that for some i, some qi ∈ IR+ and P = D−1 (qi + j6=i qjm ),
P qi − Ci (qi ) + P
X
j6=i
≤
Pm
X
j
qjm
−
X
qjm −
X
Cj (qjm )
j6=i
Cj (qjm )wP qi
− Ci (qi ) + P m
j
X
j6=i
qjm −
X
Cj (qjm )
j6=i
implying P < P m . Therefore, (P, qi ) is an admissible deviation for firm i in the oligopoly
game, which entails profit P qi − Ci (qi ), so that (P m , q m ) cannot be an oligopolistic equilibrium
outcome.
Notice that coincidence of Cournot and collusive outcomes, impossible under differentiability
of the cost and demand functions (except if there is only one active firm6 , cannot be generally
excluded. If, for instance, demand is non-differentiable at price P ∗ , one may have (under cost
6
The “collusive outcome” involving several active firms should be distinguished from the “monopoly equilib-
rium” where one firm is producing the optimal monopoly output and all other firms do not produce. For an
analysis of the monopoly equilibrium see Amir and Lambson (2000).
8
symmetry and n ≥ 2) the first order conditions:
1/n[−+ D(P ∗ )] < 1/[−+ D(P ∗ )] ≤ 1 − C 0 (D(P ∗ )/n)/P ∗
≤ 1/n[−− D(P ∗ )] < 1/[−− D(P ∗ )]
with P ∗ = P C = P m , and with − D(P ∗ ) and + D(P ∗ ) the left-hand and right-hand demand
elasticities at P ∗ , respectively.
2.2
Parameterization of equilibria
We shall now investigate the first order condition of the oligopolist’s program defined by (4).
For each firm i, we have to distinguish the case where, at the solution, it is active (qi∗ > 0)
and the case where it is not7 (qi∗ = 0). All equilibrium prices, by contrast, should be strictly
positive. Introducing Kuhn-Tucker multipliers (λi , νi ) ∈ IR2+ \ {0} associated with the first and
second constraints in (4), respectively, general first order conditions require, by the positivity of
p∗i and the non-negativity of qi∗ , that qi∗ − λi − νi = 0, and p∗i − Ci0 (qi∗ ) + νi /D0 (minj {p∗j }) ≤ 0
with (p∗i − Ci0 (qi∗ ) + νi /D0 (minj {p∗j }))qi∗ = 0. Therefore, if firm i is active, we get

qi∗

p∗i − Ci0 (qi∗ )


 = λi 
1
0


 + νi 
1
−1/D0 (minj {p∗j })

,
(11)
whereas, if firm i is inactive, we let λi = νi = 0.
The multiplier λi is associated with the min-pricing constraint and the multiplier νi with
the demand constraint. They can be interpreted as the shadow costs firm i would accept to
bear in order to ease the pressure coming from its competitors inside and outside the industry,
respectively. For an active firm, we define the normalized parameter θi ≡ λi /(λi +νi ) ∈ [0, 1] (for
an inactive firm i, θi is undefined). This parameter may be viewed as an index of the competitive
toughness displayed by firm i within the industry, at equilibrium (p∗ , q ∗ ). The corresponding
degree of monopoly of each firm i in the set of the n∗ active firms (with p∗i = minj {p∗j } and
7
Shubik (1959) suggests to call such a firm a “firm)in-being” (by analogy to the famous term “fleet in being,”
introduced by Lord Torrington and used by Kipling).
9
qi∗ > 0) can then be expressed as a function of θi :
P
qi∗ / j qj∗
p∗i − Ci0 (qi∗ )
∗
∗
= (1 − θi )
≡ µi θi , min{pj }, q ,
j
p∗i
−D(minj {p∗j })
(12)
where is the elasticity operator (i.e., f (x) ≡ (df (x)/dx)(x/f (x)), for a differentiable function
f (x)).
Oligopolistic equilibria may hence be characterized by first order conditions (12), together
with equality (5), giving a system of n∗ + 1 equations
µi (θi , P ∗ , q ∗ ) =
X
P ∗ − Ci0 (qi∗ )
(i = 1, · · · , n∗ ) and
qj∗ = D(P ∗ )
∗
P
(13)
j
∗
in n∗ + 1 unknowns (P ∗ , q ∗ ), parameterized by θ = (θ1 , · · · , θn∗ ) in the set Θ∗ ⊂ [0, 1]n of the
parameter values which entail a solution to the system (13) and ensure sufficiency of the first
order conditions. We thus obtain endogenously a parameterization by θ of the set of oligopolistic
equilibria.
We shall not examine here the question of existence and the sufficiency of first order conditions. Local curvature requirements should be somewhat softened by the presence of a kink in
the boundary of the admissible set. However it should be stressed that sufficient global conditions for the existence of an oligopolistic equilibrium are necessarily weaker than those ensuring
existence of standard solutions, such as Cournot or competitive equilibria, since these solutions,
as we have seen above, are enforceable as particular instances of oligopolistic equilibria. In any
particular case the difficulty is to determine the set Θ∗ . Illustrative examples are given below.
The following proposition shows that potential oligopolistic equilibria coincide at one extreme, when competitive toughness is minimal (θ = (0, · · · , 0)), with the Cournot solution and,
at the other extreme, when competitive toughness is maximal (θ = (1, · · · , 1)), with the competitive equilibrium. All other oligopolistic equilibria correspond to intermediate values of θ,
including (when marginal costs are constant) those associated with Bertrand competition.
We can now state the following
Proposition 3 Assuming that θ = (0, · · · , 0) (respectively θ = (1, · · · , 1)) belongs to Θ∗ , and
P
that, for any i, D−1 (qi + j6=i qj )qi − Ci (qi ) is quasi-concave (respectively Ci (qi ) is convex) in
10
qi , any oligopolistic equilibrium corresponding to θ = (0, · · · , 0) (respectively θ = (1, · · · , 1)) leads
to a Cournot (respectively a competitive) outcome.
Proof:
Recall that in the case of Cournot equilibrium the first order conditions (which are
sufficient by assumption) are:
P
qiC / j qjC
P C − Ci0 (qiC )
=
≡ µi (0, P C , q C ) for all i,
PC
−D(P C )
coinciding, by (12), with the first order conditions for an oligopolistic equilibrium with θ =
(0, · · · , 0) ∈ Θ∗ . With θ = (1, · · · , 1) ∈ Θ∗ , we see from (12) that each firm equalizes marginal
cost to price, so that we get the condition (assumed to be sufficient) for a competitive equilibrium.
The result follows.
In relation to this result, it is important to note that the degree of monopoly equations
(12) are in fact the rewriting of the behavioral equations used in the empirical literature to
estimate firm and industry conduct (how firms et prices and quantities). These equations are
typically parameterized, for each firm, by an “index of the competitiveness of oligopoly conduct”
(Bresnahan, 1989, p. 1016) which is the exact complement of our parameter θi (i.e. equal to
(1 − θi )). This parameterization is used by econometricians to nest existing theories of oligopoly.
These parameters are also directly related to the conjectural variation approach (Bowley,
1924). They can be taken as continuous-valued and used to estimate conjectural derivatives.
In spite of its lack of theoretical foundations,8 the conjectural variation approach is meant to
fill the gap left open by the theory. In this approach, each firm i when choosing its quantity is
supposed to make a specified type of conjecture concerning the reaction of the other firms to any
of its deviations. These conjectures are introduced directly into the first order conditions. They
are not part of the description of the oligopoly game. Following the presentation in Dixit (1986),
P
a sufficient specification9 consists in introducing conjectural derivatives ri = j6=i ∂qj /∂qi for
each i. The corresponding first order conditions are:
P
qi∗ / j qj∗
P ∗ − Ci0 (qi∗ )
= (1 + ri )
,
P∗
−D(P ∗ )
8
9
It has also been criticized as leading to a mismeasurement of market power (Corts, 1999).
P
Dixit considers the more general case where ri is a function of both qi and j6=i qj .
11
(14)
giving the same characterization as (12) with ri = −θi . In other words, comparing first order
conditions, the set of oligopolistic equilibrium outcomes appears as the selected subset of outcomes obtained by conjectural variations restricted to the compensating (non-collusive), i.e. ri
to be in the interval10 [−1, 0], for every i. The concept of oligopolistic equilibrium thus provides
some game-theoretic foundation to the concept of conjectural variations, since the conjectural
variation terms (within the relevant class) can be identified with the parameterization of the
equilibria of a fully specified game.
3
Other game-theoretic approaches
The formal concept of oligopolistic equilibrium has been defined as the characterization of the
Nash equilibrium of a Cournot-Bertrand oligopoly game. However, we shall now see that there
are other theoretical ways (and these are illustrative of many others) to arrive at the same
formal characterization. Each of these different theories might be more adapted to some specific
industries and some particular situations. This is why the concept of oligopolistic equilibrium
as defined by Lemma 1 has to be seen as a canonical representation and a convenient theoretical
tool.
3.1
The min-pricing scheme approach
One can define an alternative oligopoly game where firms are not committed to serve all demand. We suppose instead that each firm includes a meeting competition clause (or price-match
guarantee) in its sales contracts, guaranteeing its customers that they are not paying more than
what they would to a competitor, so that each customer acts as if facing the single market price
P min (p) = minj {pj }, where P min is the min-pricing scheme (d’Aspremont eat al., 1991). Combining this guarantee with the assumption that each firm i brings qi to the market, we again have
that it is willing to sell its output at the discount price P = min{P min (p) , D−1 (q1 + · · · + qn )}.
10
Matching variations (ri > 0) are excluded, and in particular those leading to the collusive solution.
12
We thus get payoff functions given, for firm i, by





X
min
−1 
 qi − Ci (qi ).
Πpm
(p
,
p
,
q
,
q
)
≡
min
P
(p
,
p
,
D
q
+
q
i
−i
i
−i
i
−i
i
j
i


(15)
j6=i
We thus get an alternative game, a price-matching oligopoly game in prices and quantities, using
these payoffs. The corresponding oligopolistic equilibrium (p∗ , q ∗ ), a P min -equilibrium, is a Nash
equilibrium satisfying in addition the no-rationing restriction
X
qj∗ = D(P min (p∗ )),
(16)
j
eliminating equilibria where customers would be willing to buy more at the equilibrium price
P min (p∗ ).
Proposition 4 A 2n-tuple (p∗ , q ∗ ) is a P min -equilibrium if and only if it is an oligopolistic
equilibrium, i.e. for each firm i, (p∗i , qi∗ ) is solution to the program (4) and the market balance
restriction (5) holds.
Proof:
Suppose first that (p∗ , q ∗ ) is a P min -equilibrium (so that
∗
j qj
P
= D(minj {p∗j })), but
that, for some i, and some (pi , qi ) ∈ IR2+ , pi qi −Ci (qi ) > p∗i qi∗ −Ci (qi∗ ), with pi ≤ min{p∗−i D−1 (qi +
P
P
∗
0
0
∗
−1
∗
j6=i qj )}. The pair (pi , qi ), with pi = min{pi , p−i , D (qi +
j6=i qj )}, is then a deviation w.r.t.
the P min -equilibrium, a contradiction.
To prove the other direction, suppose now (p∗ , q ∗ ) is an oligopolistic equilibrium (so that
P
again) j qj∗ = D(minj {p∗j })), but that, for some i, some (pi , qi ) ∈ IR2+ , and p0i ≡ min{pi , p∗−i , D−1 (qi +
P
∗
0
∗ ∗
∗
0
j6=i qj )}, we have pi qi − Ci (qi ) > pi qi − Ci (qi ) ≥ 0. Then (pi , qi ) satisfies the two constraints
in the program (4) and gives higher profit to firm i, again a contradiction.
Hence, the min-pricing scheme approach is another way to get oligopolistic equilibria and
a relevant one to investigate the large number of markets where the price-match guarantee is
offered.
13
3.2
The supply function approach
Another approach assumes that firms strategies are supply functions (Grossman, 1981, and
Hart, 1982).11 A supply function equilibrium is a Nash equilibrium of a game where each firm i
strategies are functions Si associating with every price pi in [0, ∞) a quantity qi = Si (pi ). The
functions Si may be restricted to some admissible set S. For any n-tuple S of supply functions
in Sn , the price P̂ (S) is said to be well defined if it is non-negative and uniquely solves the
equation
n
X
Sj (P ) = D (P ) .
(17)
j=1
The corresponding payoffs are defined by:
ΠSi (Si , S−i = P̂ (Si , S−i )Si (P̂ (Si , S−i )) − Ci (Si (P̂ (Si , S−i ))), if P̂ is well defined,
ΠSi (Si , S−i = 0,
(18)
otherwise.
∗ ) amounts to select P ∗
Observe that at an equilibrium S ∗ , for any firm i, maximizing ΠSi (Si , S−i
in
∗
∗
arg max {P Di∗ (P, S−i
) − Ci (Di∗ (P, S−i
))},
P ∈IR+
∗ ) = max{D (P ) −
with Di∗ (P, S−i
∗
j6=i Sj (P ) , 0}
P
(19)
the residual demand function. Indeed, firm
∗ ) has the unique
i could as well choose any supply function Si for which Si (P ) = Di∗ (P, S−i
solution P ∗ .
The multiplicity of supply function equilibria is well known. In order to compare this concept
with our own, we shall restrict strategies to the set S+ of non-decreasing supply functions.12 We
then have the following characterization:
Proposition 5 For any supply function equilibrium S ∗ ∈ Sn+ , there is an oligopolistic equilibrium (p∗ , q ∗ ) such that q ∗ = S ∗ (P̂ (S ∗ )) and, for any j, p∗j = P̂ (S ∗ ). Conversely, for any
oligopolistic equilibrium (p∗ , q ∗ ), there is a supply function equilibrium S ∗ ∈ Sn+ such that
S ∗ (minj {p∗j }) = q ∗ .
11
12
See also Singh and Vives (1984).
As Delgado and Moreno (2004) do. However they assume in addition that firms are identical.
14
Proof:
Let S ∗ ∈ Sn+ be a supply function equilibrium. We show that (p∗ , q ∗ ) with q ∗ =
S ∗ (P̂ (S ∗ )) and, for any j, p∗j = P̂ (S ∗ ), is an oligopolistic equilibrium. Using, for any i, the
∗ ) is decreasing in P and that the profit p q − C (q ) is
fact that the residual demand Di∗ (P, S−i
i i
i i
increasing in pi we get, by (19), that (p∗i , qi∗ ) maximizes pi qi − Ci (qi ) on
∗
∆i ≡ {(pi , qi ) ∈ IR2+ | qi ≤ Di∗ (pi , S−i
}.
By Lemma 1, for (∗ , q ∗ ) to be an oligopolistic equilibrium, (p∗i , qi∗ ) should maximize pi qi − Ci (qi )
on






X
ˆ i ≡ (pi , qi ) ∈ IR2+ | pi ≤ min(p∗j ), qi ≤ max D(pi ) −
∆
qj∗ , 0
,



j6=i
j6=i
ˆ i and ∆
ˆ i ⊂ ∆i , this is clearly true.
for every i. Since, by construction, (p∗ , q ∗ ) ∈ ∆
To prove the converse, let us suppose that (p∗ , q ∗ ) is an oligopolistic equilibrium. We may
construct an associated supply function equilibrium by imposing to every firm i an admissible
supply function Si simply characterized by a price-quantity pair (pi , qi ), and such that Si (P ) = qi
if P ≤ pi , and Si (P ) = ∞ otherwise. Clearly, the solution to (19) cannot be larger than
minj6=i {p∗j }, hence any profitable deviation by some firm i from Si∗ must involve a price below
P
minj6=i {p∗j } and a quantity below D(pi ) − j6=i qj∗ , and thus constitute a deviation with respect
to the oligopolistic equilibrium. The result follows.
We see that, with the restrictions imposed on the admissible class of supply functions, the
outcomes corresponding to the two sets of equilibrium outcomes coincide, including the Cournot
and the competitive outcomes.
To establish more clearly the relation between the two concepts, we may consider the first
order condition to firm i program (19) at equilibrium, while restricting to differentiable supply
∗ ), we get:
function in SS + . Denoting qi∗ = Di∗ (P ∗ , S−i


X 0
qi∗ + D0 (P ∗ ) −
Sj∗ (P ∗ ) (P ∗ − Ci0 (qi∗ )) = 0,
(20)
j6=i
which is equivalent to
P
qi∗ / k qk∗
P ∗ − Ci0 (qi∗ )
P
P
=
= µi (θi , P ∗ , q ∗ ),
P∗
−D(P ∗ ) + j6=i (qj∗ / k qk∗ ) ∈ Sj∗ (P ∗ )
15
(21)
with µi as defined in (12), taking
θi = 1 −
−D(P ∗ )
+
−D(P ∗ )
P ∗
.
∗
∗
∗
j6=i (qj /
k qk ) ∈ Sj (P )
P
(22)
Not surprisingly, the Cournot solution corresponds to an elasticity Sj∗ (P ∗ ) of the supply functions equal to 0 for all j, and the competitive solution to Sj∗ (P ∗ ) = ∞ for at least two j’s.
Notice also that varying the elasticities of the supply functions in the relevant class allows us
to cover the whole range of admissible values of the θi . The competitive toughness of firm i, as
measured by θi at the oligopolistic equilibrium, is seen to correspond positively to a measure
of the “reactivity of the other firms” (with respect to prices) as anticipated by firm i at the
supply function equilibrium. Although the elasticity of a supply function chosen by any firm
is indifferent from the point of view of the firm itself (for which only the price-quantity pair
actually implemented matters), it is crucial in shaping the anticipations of its competitors.
4
Two illustrative cases
In order to illustrate the concept of oligopolistic equilibrium, and visualize the set of equilibrium
outcomes as parameterized by θ ∈ Θ∗ , we refer to two standard cases of homogeneous duopoly,
the cases of linear and isoelastic demand with linear cost functions. Since technological lead
is an important feature in many applied models (for example those analyzing the relationship
between innovation and competition), he will admit asymmetric costs and assume that each firm
i has a constant marginal cost ci (by convention, c1 ≤ c2 ).
4.1
Linear demand
In the first case, we suppose demand D (P ) to be linear, equal to a − P , so that Marshallian
elasticity is P/D (P ). We assume ac2 > c2 −c1 , so that the monopoly price P M = (a+c1 )/2 that
would be set by firm 1 is larger than the cost c2 of firm 2. The case where only the efficient firm
is active at equilibrium is simple. The set of equilibria is then described by all prices between
the competitive price P W = c1 and the highest Bertrand price13 P B = c2 . When both firms
13
See footnote 6.
16
are active, from condition (12), equilibrium (p∗ , q ∗ ) with p∗1 = p∗2 = P ∗ is seen to verify for each
firm i
P ∗ − ci
qi∗ D(P ∗ )
qi∗
=
(1
−
θ
)
=
(1
−
θ
)
,
i
i
P∗
q1∗ + q2∗ P ∗
P∗
(23)
giving a market share for firm 1
q1∗
1 P ∗ − c1
q1∗
1
=
=1−
P ∗ − c2 a − P ∗ ,
∗
+ q2
1 − θ1 a − P ∗
1 − θ2
(24)
with
P∗ =
(1 − θ1 )(1 − θ2 )a + (1 − θ2 )c1 + (1 − θ1 )c2
(1 − θ1 )(1 − θ2 ) + (1 − θ2 ) + (1 − θ1 )
(25)
The equilibrium price P ∗ is lower bounded by the highest Bertrand price P B = c2 , and upper
bounded by the Cournot price P C = (a + c1 + c2 )/3, corresponding to θ2 = 1 and to θ1 = θ2 = 0,
respectively. These expressions are valid only for values of θ in [0, 1 − (c2 − c1 )/(a − c2 )] × [0, 1].
Indeed, for values of θ1 > 1 − (c2 − c1 )/(a − c2 ), firm 2 becomes inactive and we are back to the
first case.
To illustrate, we can take the values for the costs c1 = 2/3 and c2 = 1 and, choosing a = 10/3,
so that P ∗ ∈ [1, 5/3], we obtain the following representation of the set of equilibrium outcomes
with the two firms active, in the market price-market share space (P ∗ , q1∗ /(q1∗ + q2∗ )). This set
is given by the region bounded by the vertical axis and the two thick curves, the lower one
corresponding to θ1 = 0, the upper one to θ2 = 0. The increasing curves (including the thick
one) are three elements of the family of curves stemming from firm 1 first order condition (with
θ1 = 1/3, 1/6, 0, respectively, from left to right). Similarly, the decreasing concave curves are
three elements, corresponding to the same values of θ2 , of the family of curves stemming from
firm 2 first order condition.
The decreasing convex gray curve in Figure 1 corresponds to the uniform case θ1 = θ2 = θ.
It links the Cournot outcome (determined at the point of intersection of the two thick curves,
with θ = 0) to the Bertrand outcome (associated with the highest possible market share for the
efficient firm and the highest value of θ, 1 − (c2 − c1 )/(a − c2 ) = 6/7). We observe that, as θ
increases, the market expands and splits more and more asymmetrically in favor of the efficient
firm.
17
Figure 1
4.2
Isoelastic demand
The demand function is now taken to be D (P ) = P −σ , with 1/σ ∈ (1 − c1 /c2 , 2), so that
existence of a Cournot equilibrium is ensured, and firm 1 monopoly price P M = c1 /(1 − 1/σ) is
larger than the cost c2 of firm 2. From condition (12) we see that, at equilibrium (p∗ , q ∗ ) with
p∗1 = p∗2 = P ∗ , the degree of monopoly of firm i is
1 − θi qi∗
P ∗ − ci
=
,
P∗
σ q1∗ + q2∗
(26)
giving the market share form firm 1
σ P ∗ − c1
σ P ∗ − c2
q1∗
=
=
1
−
,
q1∗ + q2∗
1 − θ1 P ∗
1 − θ2 P ∗
(27)
with
P∗ =
(1 − θ2 )c1 + (1 − θ1 )c2
.
1 − θ2 ) + (1 − θ1 ) − (1 − θ1 )(1 − θ2 )/σ
(28)
The equilibrium price P ∗ is lower bounded by the highest Bertrand price P B = c2 , and upper
bounded by the Cournot price P C = (c1 + c2 )/(2 − 1/σ). It is decreasing in each θi . These
expressions are valid only for values of θ in [0, 1 − σ(1 − c1 /c2 )] × [0, 1]. For values of θ1 in
[1 − σ(1 − c1 /c2 ), 1], firm 2 becomes inactive.
18
Figure 2 shows, for parameter values σ = 1, c1 = 2/3 and c2 = 1, the set of equilibrium
outcomes with both firms active, in the market price-market share space (P ∗ , q1∗ /(q1∗ + q2∗ )) as
the region bounded by the vertical axis and the two thick curves. The solid curves are associated
with constant values of competitive toughness, corresponding to the same values of θi above:
1/3, 1/6, 0, respectively, from left to right. These curves are increasing for firm 1 and decreasing
for firm 2. The gray curve is associated with varying equal degrees of competitive toughness for
both firms. Figure 2 is strikingly similar to Figure 1, and leads to the same comments.
Figure 2
Some significant properties concerning the effects of competitive toughness on profits can be
deduced from both examples. In the uniform case (θ1 = θ2 = θ):14
• the inefficient firm looses as competitive toughness increases (both its market share and
the price decrease);
• if the cost advantage of the efficient firm is large enough, its profit increases as competitive
toughness increases (a higher market share more than compensates a lower price);
14
These properties correspond to properties (b), (c) and (d) in Definition 2.2 of Boone (2001), where the
“intensity of competition” parameter is supposed to be the same for all firms. Cournot and Bertrand regimes, but
no the intermediate cases, are considered in an example. In Boone (2000), the parameter measuring “competitive
pressure” is differentiated but his examples concentrate on comparative statics when the fundamentals vary.
19
• if costs are equal for both firms (implying equal market shares), profits decrease as competitive toughness increases.
In the case where competitive toughness is asymmetric, effects are more intricate. If the value
of θi is ket unchanged and θj , j 6= i, increases, the price as well as firm i market share fall,
implying a lower profit for firm i. However, an increase in the value of θi while keeping θj , j 6= i,
constant, is associated with a larger market share for firm i but with a lower price, so that an
increase in its profit is not guaranteed.
5
Competitive toughness under firm control
In the three justifications of an oligopolistic equilibrium, which have been given and are based
respectively on the Cournot-Bertrand, the min-pricing and the supply-function oligopoly games,
the parameters measuring the competitive toughness of firms were part of the characterization
(through first order conditions) of the set of potential equilibria. For these theories, competitive
toughness is an endogenous parameter out of the direct control of individual firms. Apart
from an empirical estimation, and in order to get a theoretical specification of these parameters,
either a more complete description of the context or an equilibrium selection argument is needed.
Examples of such possibilities are respectively given by Kreps and Scheinkman (1983) two-stage
game, with capacity recommitment followed by price competition in a symmetric duopoly, and
Delgado and Moreno (2004), who look for a coalition-proof supply function equilibrium. In both
cases the Cournot outcome is obtained, corresponding to θ ≡ 0 in the present canonical model.
But one cannot exclude the possibility of oligopoly games where the toughness parameters can
be exogenously fixed and could even be under firm control. In support of this claim, we give
two different illustrations.
5.1
Exogenous competitive parameters
Let us consider two very different interpretations of the competitive toughness parameters, taken
as exogenous. In one interpretation more competition results from a more socially-oriented
20
attitude, in the other it results from a more aggressive conduct.
5.1.1
Game 1
The first interpretation is based on a modified oligopoly game where each firm i, simply choosing
a quantity as in the Cournot model, has a modified objective function where the parameter θi ∈
[0, 1] explicitly appears and can be interpreted as a coefficient of collective concern giving some
positive weight to the total surplus. This coefficient is similar to the “coefficient of cooperation”
used in the literature (e.g. Cyert and De Groot, 1973, Martin, 2002) and inspired by Edgeworth’s
“coefficient of effective sympathy.” But here it applies not only to the sum of the other firms
profits but also to the consumers’ surplus. The profit of firm i is defined as




X
−1 q +

Πcc
qj  − Ci (qi )
i
i (qi , q−i , θi ) = (1 − θi ) qi D
j6=i

Z

+ θi
P
qi + j6=i qj
0

D−1 (Q)d Q −
X
Cj (qj ) .
j
When θi = 0, Πcc
i (qi , q−i , 0) reduces to the Cournot profit function.
When every θi = 1,
Πcc
i (qi , q−i , 1) amounts to the total surplus (total profit plus consumer surplus). Considering
an equilibrium q ∗ of this modified game, the first order conditions can be computed for n∗ active
firms i = 1, 2, · · · , n∗ :
0=
∗
∂Πcc
i (q , θi )
∂qi

=D
−1 
"
X
j
qj∗ 
#
P
(1 − θi )qi∗ / j qj∗
P
− Ci0 (q ∗ − i),
1−
−D(D−1 ( j qj∗ ))
and are equivalent to the system defined by (12) and (13), leading to the same set of potential
solutions.
5.1.2
Game 2
To introduce the second interpretation, we define a game of “tempered” Bertrand-Edgeworth
competition. To simplify, we suppose two firms with linear costs, Ci (qi ) = ci qi , with ci > 0
(i ∈ {1, 2}). The parameter θi ∈ [0, 1), fixed in advance, is interpreted as the probability that
21
firm i will have an aggressive rather than a compromising conduct. We suppose that the game
has two stages and we look for a sub game perfect equilibrium. At the first stage, each firm i
quotes the maximum price pi ∈ (0, P ) at which it commits to sell up to quantity qi ∈ [0, ∞), to
be produced in advance. At the second stage, the conduct of each firm i is decided according to
the probability θi . Under a compromising conduct, adopted with positive probability 1 − θi , firm
i supplies at quoted price pi a quantity xi ∈ [0, qi ] which should not exceed the residual demand,
namely max[D(pi ) − qj , 0]. By contrast, under an aggressive conduct, adopted with probability
θi , it undercuts its rival by an arbitrarily low amount ε > 0 (which is taken as an exogenous
parameter), and supplies at price ψi = min{pi , pj − ε} a quantity xi ∈ [0, qi ] no larger than what
it can actually sell: D(ψi ) with a compromising competitor (or with an aggressive competitor
if ψi < ψj and max[D(ψi − qj , 0] otherwise. Clearly, since supplying xi entails no cost, one of
these bounds will always be binding at second stage equilibrium. Also, we assume that, in case
of indifference, a firm always chooses to produce the maximal quantity it can sell. However, an
equilibrium may require the consumers tone rationed and we assume “efficient rationing.”
Now consider the expected profit of firm i at the first stage, anticipating the second-stage
equilibria, and involving the four states of the world (with two aggressive or two compromising
firms, or with one aggressive and the other compromising) and their corresponding probabilities.
Take, as a first case, the expected profit ΠBE
of firm i as a function of (p, q, θ) ∈ (0, P )2 ×[0, ∞)2 ×
i
[0, 1)2 when pi ≤ pj ,
ΠBE
i (p, q, θ) = (1 − θi )pi min{qi , max[D(pi ) − qj , 0]}
+ θi (1 − θj )ψi min{qi , D(ψi )}
(29)
+ θi θj ψi min{qi , max[D(ψi ) − qj , 0]} − ci qi .
Notice that the first term in this expression, corresponding to the case i is compromising, involves
only the probability (1 − θi ) since i’s payoff is the same whether j is aggressive or compromising.
When pi > pj (implying ψi = pj − ε < ψj ) the expression for the expected profit of firm i
becomes independent of probability θj :
ΠBE
i (p, q, θ) = (1 − θi )pi min{qi , max[D(pi ) − qj , 0]}
+ θi (pj − ε) min{qi , D(pj − ε)} − ci qi .
22
(30)
Observe that, for θ1 = θ2 = 0, the program of each firm becomes:
max pi qi − ci qi s.t. pi ≤ D−1 (qi + qj ),
(pi ,qi )
so that we get again the Cournot solution. For other values of θ we have the following fact:
Proposition 6 At an equilibrium (p∗ , q ∗ ) ∈ (0, P )2 ×(0, ∞)2 of the tempered Bertrand-Edgeworth
duopoly, with θ ∈ [0, 1)2 and positive supplies of both firms in all states occurring with positive
probabilities, the following conditions necessarily hold:
(i) p∗1 = p∗2 = D−1 (q1∗ + q2∗ ) and, for i = 1, 2,
(ii) qi∗ ∈ arg maxqi ∈[0,D(p∗j −ε)−qj∗ ] π(qi , θi ), with
πi (qi , θi ) ≡ [(1 − θi )D−1 (qi + qj∗ ) + θi (p∗j − ε) − ci ]qi .
Proof:
Under condition (i), necessity of condition (ii) is easily proved. Suppose that πi (qi , θi ) >
π(qi∗ , θi ) for some qi ∈ [0, D(p∗j − ε) − qj∗ ]. Then the pair (pi , qi ) with pi = D1 (qi + qj∗ ) ∈
∗
∗
∗
[p∗j − ε, P ) is a profitable deviation for firm i, since ΠBE
i (pi , pj , qi , qj , θ) = πi (qi , θi ) > πi (qi , θi ) =
∗ ∗
ΠBE
i (p , q , θ) by Eqs. (29) and (30).
It remains to establish necessity of condition (i). Let us begin by considering the case of an
equilibrium (p∗ , q ∗ ) such that p∗1 = p∗2 = P ∗ , and show that q1∗ = q2∗ = D(P ∗ ) in this case. If
q1∗ + q2∗ < D(P ∗ ), then by Eq. (29),
∗ ∗
∗
∗
∗
ΠBE
i (p , q , θ) = [(1 − θi )P + θi (P − ε) − ci ]qi , i = 1, 2,
and we see from Eq. (30) that some firm i would increase its profit by (slightly) increasing its
price. If q1∗ +q2∗ > D(P ∗ ) then, again by Eq. (29), θi (P ∗ −ε) ≥ ci and P ∗ ∈ arg maxpi {pi [D(pi )−
qj∗ ]} for i = 1, 2. An increase by firm i of its price to pi larger than but arbitrarily close
to P ∗ would only have a second order effect on the revenue component pi [D(pi ) − qj∗ ] at its
maximum p∗i = P ∗ , but would determine a discontinuous positive increase on the complementary
revenue component (due to the shift from (29) to (30)), through a jump from (P ∗ − ε)[(1 −
23
θj ) min{qi∗ , D(P ∗ − ε)} + θj min{qi∗ , D(P ∗ − ε) − qj∗ }] to (P ∗ − ε) min{qi∗ , D(P ∗ − ε)}. Thus,
q1∗ + q2∗ = D(P ∗ ) if p∗1 = p∗ − 2 = P ∗ .
Now consider an equilibrium (p∗ , q ∗ ) such that, say, p∗i < p∗j . The are two possible cases.
Case 1. If θj (p∗i − ε) < cj , then qj∗ = D(p∗j ) − qi∗ by Eq. (30), since qj∗ < D(p∗j ) − qi∗ would
trigger an upward price deviation by firm j, and qj∗ < D(p∗j ) − qi∗ a downward quantity deviation
by the same firm. But then p∗i < p∗j = D−1 (qi∗ + qj∗ ) implies qi∗ < D(p∗i ) − qj∗ and (by Eq. (29))
firm i would gain by increasing its price. Case 1 is thus excluded.
Case 2. If θj (p∗i − ε) ≥ cj , firm j optimal choice should satisfy qj∗ = D(p∗i − ε) > D(ψi∗ ).15
Thus D(p∗i ) − qj∗ ≤ D(ψi∗ ) − qj∗ < 0 and, by Eq. (29), firm i supplies x∗i = 0 in at least one
state with positive probability (both firms compromising with probability (1 − θi )(1 − θj ) > 0),
contradicting the assumption of the proposition and leading to exclusion of case 2.
Therefore, in both cases we get a contradiction, and condition (i) follows.
By computing the first order condition for maximization of πi (qi , θi ) at an equilibrium (p∗ , q ∗ )
with θ ∈ [0, 1)2 and positive supplies of both firms in all states that occur with positive probabilities, we obtain, using p∗1 = p∗2 = P ∗ = D−1 (q1∗ + q2∗ ) and qi∗ ∈ (0, D(P ∗ − ε) − qj∗ ), for
i = 1, 2,
qi∗ /(q1∗ + q2∗ )
ε
P ∗ − ci
=
(1
−
θ
)
+ θi ∗ .
i
∗
∗
P
−D(P )
P
(31)
As ε tends to zero, this condition eventually coincides with condition(12), implying the same
set of (potential) equilibria. The case where θi = 1 for some i can be treated as limit cases.
5.2
Choosing θ strategically
Although the interpretation of θ is very different in the two examples, it is exogenously fixed
in both and the set of resulting potential equilibrium (with both firms active) are the same.
Moreover, in both examples, one can consider the situation where each firm i would choose its
θi strategically in a preliminary stage. We shall not investigate extensively this question here,
15
In case θj (p∗i − ε) = cj , firm j is indifferent between any qj ı[D(p∗j ) − qi∗ , D(p∗i − ε)], but is supposed to choose
D(p∗i − ε).
24
but we may at least show what answer we get in the two illustrative cases of linear and isoelastic
demands.
In both case, with linear costs, the profit of firm i is equal to the product of the degree of
monopoly, the market share and the aggregate expenditure level. When increasing its competitive toughness, a firm increases its market share and may or may not increase the aggregate
expenditure, but anyway must endure a loss in its degree of monopoly. In the linear demand
case, using (23) and (24), the equilibrium profit of firm i combines these three components as
follows:
∗
Πi (P , P
∗
, qi∗ , qj∗ )
= (1 − θi )
qi∗
q1∗ + q2∗
2
(a − P ∗ )2 =
(P ∗ − ci )2
.
1 − θi
(32)
The equilibrium profit of firm i is thus an increasing function of θi directly, and a decreasing
function of θi through the price P ∗ . From (25), it can be computed to be
Π∗i (θ)
= (1 − θi )
(cj − ci ) + (1 − θj )(a − ci )
(1 − θi ) + (1 − θj ) + (1 − θi )(1 − θj )
2
,
(33)
for θ ∈ [0, 1 − (c2 − c1 )/(a − c2 )] × [0, 1] (with c2 − c1 < a − c2 ). Thus, at the first stage, the
payoffs are given by (Π∗1 (θ), Π∗2 (θ)) and the strategy spaces are [0, 1 − (c2 − c1 )/(a − c2 )] and
[0, 1] respectively for firm 1 and firm 2. In order to determine the best reply of firm i in θi , it is
useful to derive the sign of the partial derivative of Π∗i (θi , θj ) with respect to θi :
sign{θi Π∗i (θ)} = sign{1 − (2 − θj )θi }.
(34)
From this formula, the quasi-concavity of Π∗i (θi , θj ) in θi is straightforwardly established since the
sign can only switch from positive (at θi = 0) to negative. Also, when the two firms have equal
unit costs, the sign can only be zero for both firms if θi = θj = 1, which leads to the Bertrand
equilibrium at the second stage. Otherwise, the efficient firm 1 will choose at equilibrium the
lowest competitive toughness that eliminates its rival, namely θ1 = 1 − (c2 − c1 )/(a − c2 ), leading
to the price P B = c2 . Therefore, in the linear demand case, the only oligopolistic equilibrium
outcome, resulting from the sub-game perfect equilibrium of the two-stage game, is the Bertrand
outcome with the highest possible equilibrium price.16
16
If individual competitive toughness is interpreted as a degree of aggressively, then this conclusion is similar
25
In the isoelastic demand case, combining the degree of monopoly (26), the market share (27)
and the aggregate expenditure P 1−σ gives the following expression for the profit function at
equilibrium:
∗
Πi (P , P
∗
, qi∗ , qj∗ )
1 − θi
=
σ
qi∗
q1∗ + q2∗
2
P
∗1−σ
σ
=
1 − θi
P ∗ − ci
P∗
2
P ∗1−σ .
(35)
From (28), the equilibrium profit of firm i can be written as a function of θ:
Π∗i (θ
cj − ci + (1 − θj )ci /σ 2
= σ(1 − θi )
(1 − θi )cj + (1 − θj )ci
1−σ
(1 − θi )cj + (1 − θj )ci
,
×
(1 − θi ) + (1 − θj ) − (1 − θi )(1 − θj )/σ
(36)
for θ ∈ [0, 1 − σ(1 − c1 /c2 )] × [0, 1] (with c1 ≤ c2 and 1/σ ∈ (1 − c1 /c2 , 2)). This again determines
both the payoffs and the strategy spaces of both firms. As above, in order to analyze the best
reply of firm i in θi , it is useful to evaluate the sign of the partial derivative of Π∗i (θi , θj ) with
respect to θi , sign{∂i Π∗i (θ)}, which can be reduced to the sign of the following expression
(σ − 1)(1 − θi )(1 − θj )[cj − (1 − (1 − θj )/σ)ci ]
(37)
+ [(1 − θi ) + (1 − θj ) − (1 − θi )(1 − θj )/σ][(1 − θi )cj − (1 − θj )ci ].
This sign is clearly positive for any θi ∈ [0, 1) if θj = 1. So taking θ = (1 − σ(1 − c1 /c2 ), 1)
together with the Bertrand equilibrium outcome (with price P B = c2 and firm 2 inactive when
c1 < c2 ) describes a sub-game perfect equilibrium of the two-stage game for any σ (with 1/σ ∈
(1 − c1 /c2 , 2)).
But other sub-game perfect equilibria exist for some values of the parameters. If σ = 1,
there exists a sub-game perfect equilibrium, with both firms active in the associated sub-game,
and with equilibrium strategies θ ∈ [0, c1 /c2 ) × [0, ] such that (1 − θ1 )/(1 − θ2 ) = c1 /c2 , so that
∂i Π∗i (θ) = 0, for i = 1, 2 (because of the upper bound imposed on θ1 , c1 /c2 > 1/2 is required).
In the symmetric case (i.e. c1 /c2 = 1) for instance, θ = 0 together with the Cournot equilibrium
outcome is a sub-game perfect equilibrium of the two stage game.
to the one obtained by Boone (2004) showing that a more aggressive outcome (here, the elimination of the less
efficient firm) should be expected when firms differ in their efficiency levels.
26
If σ 6= 1, from the two first order conditions for an interior maximum ∂1 Π∗1 (θ) = ∂2 Π∗2 (θ) = 0,
we get
cj − (1 − (1 − θj )/σ)ci = −ci + (1 − (1 − θi )/σ)cj ,
Leading to
(1 − θj )ci = −(1 − θi )cj ,
which is only possible for θ = (1, 1). But θ1 = 1 is outside firm 1 admissible strategy space
except when the two firms have identical costs (a case already treated). This excludes equilibria
with interior θ. Taking θ∗ = (0, 0) we can however easily check that ∂i Π∗i (θ∗ ) ≤ 0, for i = 1, 2,
under the condition that c1 /c2 ≥ (1 + σ − 1/σ)/σ, hence leading to the Cournot equilibrium
outcome. The condition is satisfied for any cost configuration if 1 + σ − 1/σ ≤ 0 (that is,
√
σ ≤ ( 5 − 1)/2 = 0.618). Otherwise, the condition forbids a too large efficiency gap between
the two firms, and the more so the closer σ becomes to one.
The existence of a subgame perfect equilibrium leading to the Cournot outcome under enough
complementarity ( a low enough σ) is easily explained by the fact that, in that case, among the
three terms in the firm payoff – the market share, the degree of monopoly and the aggregate
expenditure level – the last two are now decreasing in the firm competitive toughness (and the
more so the higher the level of complementarity).
6
Conclusion
In this paper we have seen that a great variety of oligopolistic regimes can be analyzed in a single
static model where all regimes that are potentially enforceable as non-cooperative equilibria can
be parameterized in terms of individual competitive toughness. Oligopolistic situations can
vary in many respects: their sectoral characteristics, the efficiency distribution of firms, the
norms of conduct in the industry, the timing of decisions, the consumer rationing scheme, the
contracting possibilities, etc. The specification of all these features may generate a complex
model in which the competition regime is well determined. An example already mentioned
is Kreps and Scheinkman (1983). But our claim is that, in many oligopolistic situations, the
27
equilibrium analysis can be performed in a single reduced canonical model. This canonical model
then provides a convenient tool to investigate some relevant issues for competition and economic
policy.17 Moreover it is ready-made for empirical testing, since its parameterization happens to
be isomorphic to the one used in econometric models adopting the “conduct parameter method.”
Of course, many issues remain to be treated. The extension of the concept to differentiated
good industries has already been undertaken (d’Aspremont and Dos Santos Ferreira, 2005, and
d’Aspremont et al, 2007). Furthermore, the existence problem requires more development,
in particular to explore the role of the kink in the producer’s feasibility frontier in relieving
curvature conditions. In studying the existence problem though, there is now a crucial change
in focus, since the main question is not the existence of equilibrium in a particular regime but
the determination of the set of enforceable competition regimes, a set that changes with the
oligopolistic situation. The study of the variation of this set in relation to the context is a new
and important issue on the agenda.
References
Alger, D.R., 1979. Markets where firms select both prices and quantities. An essay on the
foundations of microeconomic theory. Doctoral Dissertation, Northwestern University.
Amir, R., Lambson, V.E., 2000. On the effects of entry in Cournot markets. Rev. Econ. Stud.
67, 235–254.
d’Aspremont, C., Dos Santos Ferreira, R., 2005. Oligopolistic competition as a common agency
game. CORE DP 2005/18, Université catholique de Louvain.
d’Aspremont, C., Dos Santos Ferreira, R., Gérard-Varet, L.-A., 1991. Pricing schemes and
Cournotian equilibria. Amer. Econ. Rev. 81, 666–672.
17
See, for example, d’Aspremont et al. (2004) where this approach is used to study the relationship between
competitive toughness and the incentives to R& D-investment and growth.
28
d’Aspremont, C., Dos Santos Ferreira, R., Gérard-Varet, L.-A., 2004. Strategic R& D investment, competitive toughness and growth. CORE DP 2004/14, Université catholique de Louvain.
d’Aspremont, C., Dos Santos Ferreira, R., Gérard-Varet, L.-A., 2007. Competition for market
share or for market size: Oligopolistic equilibria with varying competitive toughness. Int. Econ.
Rev. 48, 761–784.
Bertrand, J., 1883. Review of Walras’ Théorie mathématique de la richesse sociale and Cournot’s
Recherches sur les principes mathématiques de la théorie des richesses. J. Savants, 499-508. English transl. by Friedman, J.W. In: Daughety, A.F. (Ed.), Cournot Oligopoly, Characterization
and Applications. Cambridge University Press, Cambridge, 1988, pp. 73–81.
Blume, A., 2003. Bertrand without fudge. Econ. Letters 78, 167–168.
Boone, J., 2000. Competitive pressure: The effects on investments in product and process
innovation. RAND J. Econ. 31, 549–569.
Boone, J., 2001. Intensity of competition and the incentive to innovate. Int. J. Ind. Organ. 19,
705–726.
Boone, J., 2004. Balance of power, CEPR DP 4733.
Bowley, A., 1924. The Mathematical Groundwork of the Economics. Oxford University Press,
Oxford.
Bresnahan, T., 1989. Empirical studies of industries with market power. In: Schmalensee, R.,
Willig, R.D. (Eds.), Handbook of Industrial Organization, vol. II. Elsevier Science Publishers
B.V., pp. 1011–1057.
Cournot, A., 1838. Recherches sur les principes mathématiques de la théorie des richesses.
Hachette, Paris. English transl. by Bacon, N.T. Hafner Publishing Company, Ltd., London,
1960. First edition 1897.
29
Corts, K.S., 1999. Conduct parameters and the measurement of market power. J. Econometrics
88, 227–250.
Cyert, R.M., DeGroot, M.H., 1973. An analysis of cooperation and learning in a duopoly
context. Amer. Econ. Rev. 63, 24–37.
Delgado, J., Moreno, D., 2004. Coalition-proof supply function equilibria in oligopoly. J. Econ.
Theory 114, 231–234.
Dixit, A.K., 1986. Comparative statics for oligopoly. Int. Econ. Rev., 107–122.
Friedman, J.W., 1988. On the strategic importance of prices versus quantities. RAND J. Econ.
19, 607–622.
Green, R.J., Newbery, D.M., 1992. COmpetition in the British electricity spot market. J. Polit.
Economy 100, 929–953.
Grossman, S., 1981. Nash equilibrium and the industrial organization of markets with large
fixed costs. Econometrica 49, 1149–1172.
Kreps, D., Scheinkman, J.A., 1983. Quantity precommitment and Bertrand competition yield
Cournot outcomes. Bell J. Econ. 14, 326–337.
Hart, O., 1982. Reasonable conjectures. DP Suntory Toyota International Centre for Economics
and Related Disciplines, London School of Economics.
Martin, S., 2002. Advanced Industrial Economics, 2nd edition. Blackwell Publishers, Oxford.
Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University
Press, New York.
Maskin, E., 1986. Oligopolistic markets with price-setting firms. In: AEA Papers and Proceedings. Amer. Econ. Rev. 76, 382–386.
30
Salop, S.C., 1986. Practices that (credibly) facilitate oligopoly coordination. In: Stiglitz, J.E.,
Mathewson, G.F. (Eds.), New Developments in the Analysis of Market Structure. Macmillan
Press, London, pp. 265–290.
Shubik, M., 1959. Strategy and Market Structure. Wiley, New York.
Shubik, M., Levitan, R., 1980. Market Structure and Behavior. Harvard University Press,
Cambridge.
Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND
J. Econ. 15, 546–554.
Vives, X., 1999. Oligopoly Pricing: Old Ideas and New Tools. MIT Press, Cambridge, MA.
31